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| Mirrors > Home > MPE Home > Th. List > fmptapd | Structured version Visualization version GIF version | ||
| Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.) |
| Ref | Expression |
|---|---|
| fmptapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| fmptapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| fmptapd.s | ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) |
| fmptapd.c | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) |
| Ref | Expression |
|---|---|
| fmptapd | ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fmptapd.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵) | |
| 2 | fmptapd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 3 | fmptapd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 4 | 1, 2, 3 | fmptsnd 7189 | . . 3 ⊢ (𝜑 → {〈𝐴, 𝐵〉} = (𝑥 ∈ {𝐴} ↦ 𝐶)) |
| 5 | 4 | uneq2d 4168 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
| 6 | mptun 6714 | . . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)) | |
| 7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))) |
| 8 | fmptapd.s | . . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆) | |
| 9 | 8 | mpteq1d 5237 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
| 10 | 5, 7, 9 | 3eqtr2d 2783 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {〈𝐴, 𝐵〉}) = (𝑥 ∈ 𝑆 ↦ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 {csn 4626 〈cop 4632 ↦ cmpt 5225 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-mpt 5226 |
| This theorem is referenced by: fmptpr 7192 poimirlem3 37630 poimirlem4 37631 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 |
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